Columnar competitive model for solving multi-traveling salesman problem

This paper studies an optimization problem: multi-traveling salesman problem (MTSP), which is an extension of the well known TSP. A columnar competitive model (CCM) of neural networks incorporates with a winner-take-all learning rule is employed to solve the MTSP. Stability conditions of CCM for MTSP is exploited by mathematical analysis. Parameters settings of the network for guaranteeing the network converges to valid solutions are discussed in detail. Simulations are carried out to illustrate the performance of the columnar competitive model compare to the heuristic algorithms: Tabu Search. 2005 Elsevier Ltd. All rights reserved.

[1]  Nicos Christofides,et al.  An Algorithm for the Vehicle-dispatching Problem , 1969 .

[2]  Andrew B. Whinston,et al.  Computer-Assisted School Bus Scheduling , 1972 .

[3]  J. Svestka,et al.  Computational Experience with an M-Salesman Traveling Salesman Algorithm , 1973 .

[4]  C. S. Orloff Routing a fleet of M vehicles to/from a central facility , 1974, Networks.

[5]  Chul E. Kim,et al.  Approximation algorithms for some routing problems , 1976, 17th Annual Symposium on Foundations of Computer Science (sfcs 1976).

[6]  John J. Hopfield,et al.  Simple 'neural' optimization networks: An A/D converter, signal decision circuit, and a linear programming circuit , 1986 .

[7]  Sanjit K. Mitra,et al.  Alternative networks for solving the traveling salesman problem and the list-matching problem , 1988, IEEE 1988 International Conference on Neural Networks.

[8]  Behrooz Kamgar-Parsi,et al.  On problem solving with Hopfield neural networks , 1990, International 1989 Joint Conference on Neural Networks.

[9]  Jean-Yves Potvin,et al.  A Generalized K-Opt Exchange Procedure For The MTSP , 1989 .

[10]  Mahesan Niranjan,et al.  A theoretical investigation into the performance of the Hopfield model , 1990, IEEE Trans. Neural Networks.

[11]  Shigeo Abe Global convergence and suppression of spurious states of the Hopfield neural networks , 1991, [Proceedings] 1991 IEEE International Joint Conference on Neural Networks.

[12]  Hiroshi Nozawa,et al.  A neural network model as a globally coupled map and applications based on chaos. , 1992, Chaos.

[13]  Martin W. P. Savelsbergh,et al.  The General Pickup and Delivery Problem , 1995, Transp. Sci..

[14]  Kazuyuki Aihara,et al.  Chaotic simulated annealing by a neural network model with transient chaos , 1995, Neural Networks.

[15]  Michel Gendreau,et al.  The m-Traveling Salesman Problem with Minmax Objective , 1995, Transp. Sci..

[16]  Hidetomo Ichihashi,et al.  A simple steepest descent method for minimizing Hopfield energy to obtain optimal solution of the TSP with reasonable certainty , 1995, Proceedings of ICNN'95 - International Conference on Neural Networks.

[17]  Mengkang Peng,et al.  An Investigation into the Improvement of Local Minima of the Hopfield Network , 1996, Neural Networks.

[18]  Kate Smith-Miles An argument for abandoning the travelling salesman problem as a neural-network benchmark , 1996, IEEE Trans. Neural Networks.

[19]  M. Pirlot General local search methods , 1996 .

[20]  Kate Smith-Miles,et al.  On chaotic simulated annealing , 1998, IEEE Trans. Neural Networks.

[21]  César Rego,et al.  Relaxed tours and path ejections for the traveling salesman problem , 1998, Eur. J. Oper. Res..

[22]  Takao Enkawa,et al.  Competition-based neural network for the multiple travelling salesmen problem with minmax objective , 1999, Comput. Oper. Res..

[23]  Pheng-Ann Heng,et al.  Winner-take-all discrete recurrent neural networks , 2000 .

[24]  Sadiq M. Sait,et al.  Evolutionary algorithms, simulated annealing and tabu search: a comparative study , 2001 .

[25]  Pedro M. Talaván,et al.  Parameter setting of the Hopfield network applied to TSP , 2002, Neural Networks.

[26]  Yoshiyasu Takefuji,et al.  An artificial maximum neural network: a winner-take-all neuron model forcing the state of the system in a solution domain , 2004, Biological Cybernetics.

[27]  J. J. Hopfield,et al.  “Neural” computation of decisions in optimization problems , 1985, Biological Cybernetics.

[28]  G. Pawley,et al.  On the stability of the Travelling Salesman Problem algorithm of Hopfield and Tank , 2004, Biological Cybernetics.

[29]  Zhang Yi,et al.  A columnar competitive model for solving combinatorial optimization problems , 2004, IEEE Transactions on Neural Networks.

[30]  A. d’Onofrio Fractal growth of tumors and other cellular populations: Linking the mechanistic to the phenomenological modeling and vice versa , 2009, 1309.3329.